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Minimisers of the Allen-Cahn equation and the asymptotic plateau problem on hyperbolic groups. (English) Zbl 1420.35109

Summary: We investigate the existence of non-constant uniformly-bounded minimal solutions of the Allen-Cahn equation on a Gromov-hyperbolic group. We show that whenever the Laplace term in the Allen-Cahn equation is small enough, there exist minimal solutions satisfying a large class of prescribed asymptotic behaviours. For a phase field model on a hyperbolic group, such solutions describe phase transitions that asymptotically converge towards prescribed phases, given by asymptotic directions. In the spirit of de Giorgi’s conjecture, we then fix an asymptotic behaviour and let the Laplace term go to zero. In the limit, we obtain a solution to a corresponding asymptotic Plateau problem by \(\Gamma\)-convergence.

MSC:

35J99 Elliptic equations and elliptic systems
30L99 Analysis on metric spaces
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